Automorphism Groups And Fixing Number Of Several Classes Of Graphs Associated To An Algebraic Object

The automorphism group of a graph reflects the symmetry of the graph,while the fixing number and metric dimension of a graph are two parameters which 'destroy' the automorphism and symmetry.Generally,it is an important however a difficult work to describe the full automorphism group both in graph theory and in algebra.In this paper,we focus on the automorphism groups and fixing numbers of three classes of algebraic graphs(i.e.,the inclusion graph on a finite group,the zero-divisor graph on a ring,the rank-decreasing graph on a matrix semigroup),and sometimes compute the metric dimension of the corresponding graph.There are five sections in the paper:The first section is an introduction.It mainly includes three aspects:1)We intro-duce the research background on automorphisms and fixing number of a graph associated with an algebraic object;2)We present the research status of automorphisms and fixing number of three classes of graphs(that is inclusion graphs defined on finite groups,zero-divisor graphs defined on rings,and rank-decreasing graphs defined on matrix rings);3)We provide some basic definitions which will be used later.In the second section,we study some properties of the inclusion graphs over finite nilpotent groups.Mainly includes:1)We determine the finite groups whose inclusion graph is complete or edgeless;2)By characterizing the independent dominating sets of inclusion graph In(Cn),we describe the automorphism group of In(Cn),and as an appli-cation of automorphisms,we compute the fixing number of In(Cn),where Cn is a finite cyclic group of order n;3)We discuss the diameter,perfectness and planarity of inclusion graphs over a finite nilpotent group.The results on diameter and planarity extend those introduced by Devi and Rajkumar.In the third section,we consider the automorphism group of the directed zero-divisor graph,which allows loops at some vertices,of a ring.Mainly includes:1)We characterize the automorphism group of directed zero-divisor graph of a finite semisimple ring;2)We give a deep description for the automorphism group of directed zero-divisor graph of a finite semisimple ring.The results of this part generalize those obtained by Wang et al,and Zhou et al.In the forth section,we study the fixing number and metric dimension of undirected zero-divisor graphs over three classes of rings.More precisely,we calculate the fixing number for undirected zero-divisor graphs on ?i=1n Fi(where Fi is a finite field),and the fixing number as well as metric dimension for undirected zero-divisor graphs on Zn or a full matrix ring over a finite field,respectively.Moreover,we determine when the undirected zero-divisor graphs of these rings are FED-graphs.In the fifth section,we depict the automorphism group of the rank-decreasing graph on a full matrix semigroup over a finite field.The result spread that obtained by Wang et al.